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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Base-invariance implies Benford’s law

Author: Theodore P. Hill
Journal: Proc. Amer. Math. Soc. 123 (1995), 887-895
MSC: Primary 60A10; Secondary 28D05
MathSciNet review: 1233974
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Abstract: A derivation of Benford’s Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law. The only base-invariant distributions are shown to be convex combinations of two extremal probabilities, one corresponding to point mass and the other a log-Lebesgue measure. The main tools in the proof are identification of an appropriate mantissa $\sigma$-algebra on the positive reals, and results for invariant measures on the circle.

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Keywords: First-digit problem, base-invariance, scale-invariance, Benford’s Law, invariant measure, <I>n</I>th digit law
Article copyright: © Copyright 1995 American Mathematical Society